Distinguish mutually exclusive events from independent events
Identify events with no shared outcome · Explain why disjoint positive-probability events are dependent · Check a case where overlapping events are independent
Exam scope and assessment notes
Extended content; not a Core requirement. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Addition and product rules
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Identify events with no shared outcome
- Explain why disjoint positive-probability events are dependent
- Check a case where overlapping events are independent
Two different relationships
Mutually exclusive events cannot occur together, so their joint probability is zero. Independent events have joint probability equal to the marginal product. If two disjoint events each have positive probability, their product is positive while their joint is zero; they are not independent. Knowing one occurred excludes the other.
A zero-probability event is an exception to the positive-probability statement: its joint with any event is zero, matching a zero product. Avoid the blanket assertion that mutually exclusive events can never be independent.
Overlap can coexist with independence
For a fair die, even has probability 1/2 and a multiple of 3 has probability 1/3. Both means 6, with probability 1/6. The product is 1/6, so these two events are independent under this model even though they overlap. Their independence is a property of the event probabilities, not of different physical experiments.
Sources for this lesson
Pause and explain
Are disjoint events with positive probabilities independent?
Worked example
For a fair die, are even and odd independent?
Show the worked solution
- Each has probability 1/2, but they cannot occur together.
- The joint 0 differs from the product 1/4.
Answer Mutually exclusive, and not independent.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
For a fair die, classify the events result 1 and result 2 as mutually exclusive and/or independent.
Give me a hint
Compare their joint with (1/6)(1/6).
Compare my reasoning
- They cannot occur together: joint 0.
- Their product is 1/36, so they are dependent.
Mutually exclusive; not independent.
Look for these in your work
- Identify the shared outcomes
- Apply the joint-product test
For a fair die, test whether even and multiple of 3 are independent and whether they are mutually exclusive.
Give me a hint
The common outcome is 6.
Compare my reasoning
- Joint=1/6; marginals are 1/2 and 1/3.
- Their product is 1/6, matching the joint, but 6 lies in both.
Independent; not mutually exclusive.
Look for these in your work
- Allow overlapping independent events
- Compare exact fractions
For a fair die, event A is result 7 and event B is even. Can these events be both mutually exclusive and independent?
Give me a hint
A is empty in this sample space.
Compare my reasoning
- P(A)=0 and P(A and B)=0.
- P(A)P(B)=0, so the independence condition holds as well.
Yes; the zero-probability event is the exception to the positive-probability argument.
Look for these in your work
- Keep the event within the model
- Recognise the zero-probability exception
Common mistakes
- “Cannot happen together” is not independence.
- Independence is possible for events from one experiment.
What can you explain now?
Are disjoint events with positive probabilities independent?
Compare with the explanation
No; their zero joint differs from their positive product.
Occurrence of one rules out the other.
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Make it stick
Are disjoint events with positive probabilities independent?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.